Find \(\int \frac{x^{3} + 5x + 1}{x^{3}} \mathrm {d} x\)
A. \(x^{2} + 10x + c\)
B. \(x + \frac{5}{3}x^{3} + x^{4} + c\)
C. \(x - 5x^{2} - 2x^{3} + c\)
D. \(x - \frac{5}{x} - \frac{1}{2x^{2}} + c\)
Correct Answer: D
Explanation
\(\frac{x^{3} + 5x + 1}{x^{3}} \equiv 1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}\)
\(\equiv \int (1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}) \mathrm {d} x = \int (1 + 5x^{-2} + x^{-3}) \mathrm {d} x\)
= \((x - 5x^{-1} - \frac{1}{2}x^{-2} + c)\)
= \(x - \frac{5}{x} - \frac{1}{2x^{2}} + c\).
